IV. MATERIALES Y METODOS
4.1 Metodología de campo
4.1.4 Tipo de estudio, tamaño de la muestra y recolección
These are just the usual definitions of various kinds of closed category — monoidal, braided monoidal and symmetric monoidal — written in a new style. This new style lets usbuild such categories from logical systems. To do this, we take the objects to be propositions and the morphisms to be equivalence classes of proofs, where the equivalence relation is generated by the equations listed in the definitions above.
However, the full advantages of this style only appear when we dig deeper into proof theory, and generalize the expressions we have been considering:
X`Y to ‘sequents’ like this:
X1, . . . ,Xn`Y.
Loosely, we can think of such a sequent as meaning
The advantage of sequents is that they let us use inference rules that — except for the cut rule and the identity rule — have the ‘subformula property’ mentioned near the end of Section 3.1.
Formulated in terms of these inference rules, the logic of closed symmetric monoidal categories goes by the name of ‘multiplicative intuitionistic linear logic’, or MILL for short [48, 82]. There is a ‘cut elimination’ theo- rem for MILL, which says that with a suitable choice of other inference rules, the cut rule becomes redundant: any proof that can be done with it can be done without it. This is remarkable, since the cut rule corresponds tocomposition of morphismsin a category. One consequence is that in the free symmetric monoidal closed category on any set of objects, the set of morphisms between any two objects isfinite. There is also a decision procedure to tell when two morphisms are equal. For details, see Trimble’s thesis [95] and the papers by Jay [54] and Soloviev [90]. Also see Kelly and Mac Lane’s coherence theorem for closed symmetric monoidal categories [61], and the related theorem for compact symmetric monoidal categories [62].
MILL is just one of many closely related systems of logic. Most include extra features, but somesubtract features. Here are just a few examples:
• Algebraic theories. In his famous thesis, Lawvere [69] defined an algebraic theoryto be a cartesian category where every object is ann-fold cartesian powerX× · · · ×X (n≥0) of a specific objectX. He showed how such categories regarded as logical theories of a simple sort — the sort that had previously been studied in ‘universal algebra’ [25]. This work initiated the categorical approach to logic which we have been sketching here. Crole’s book [34] gives a gentle introduction to algebraic theories as well as some richer logical systems. More generally, we can think of any cartesian category as a generalized algebraic theory.
• Intuitionistic linear logic (ILL). ILL supplements MILL with the operations familiar from intuitionistic logic, as well as an operation ! turning any proposition (or resource)Xinto an ‘indefinite stock of copies of X’. Again there is a nice category-theoretic interpretation. Bierman’s thesis [23] gives a good overview, including a proof of cut elimination for ILL and a proof of the result, originally due to Girard, that intuitionistic logic can be be embedded in ILL.
• Linear logic (LL). For full-fledged linear logic, the online review article by Di Cosmo and Miller [37] is a good place to start. For more, try the original paper by Girard [43] and the book by Troelstra [96]. Blute and Scott’s review article [24] serves as a Rosetta Stone for linear logic and category theory, and so do the lectures notes by Schalk [82].
• Intuitionistic Logic (IL). Lambek and Scott’s classic book [67] is still an excellent introduction to intu- itionistic logic and cartesian closed categories. The online review article by Moschovakis [77] contains many suggestions for further reading.
To conclude, let us say precisely what an ‘inference rule’ amounts to in the setup we have described. We have said it gives a function from a product of homsets to a homset. While true, this is not the last word on the subject. After all, instead of treating the propositions appearing in an inference rule asfixed, we can treat them asvariable. Then an inference rule is really a ‘schema’ for getting new proofs from old. How do we formalize this idea?
First we must realize thatX `Y is not just a set: it is a setdepending in a functorial wayonXandY. As noted in Definition 14, there is a functor, the ‘hom functor’
sending (X,Y) to the homset hom(X,Y)=X`Y. To look like logicians, let us write this functor as`.
Viewed in this light, most of our inference rules arenatural transformations. For example, rule (a) is a natural transformation between two functors fromCop×C3to Set, namely the functors
(W,X,Y,Z)7→W `(X⊗Y)⊗Z and
(W,X,Y,Z)7→W `X⊗(Y⊗Z).
This natural transformation turns any proof
f:W →(X⊗Y)⊗Z into the proof
aX,Y,Zf:W →X⊗(Y⊗Z).
The fact that this transformation isnaturalmeans that it changes in a systematic way as we varyW,X,YandZ. The commuting square in the definition of natural transformation, Definition 4, makes this precise.
Rules (l), (r), (b) and (c) give natural transformations in a very similar way. The (⊗) rule gives a natural transformation between two functors fromCop×C×Cop×Cto Set, namely
(W,X,Y,Z)7→(W`X) × (Y `Z) and
(W,X,Y,Z)7→W⊗Y`X⊗Z.
This natural transformation sends any element (f,g)∈hom(W,X)×hom(Y,Z) to f⊗g.
The identity and cut rules are different: theydo notgive natural transformations, because the top line of these rules has a different number of variables than the bottom line! Rule (i) says that for eachX∈Cthere is a function
iX: 1 → X`X
picking out the identity morphism 1X.What would it mean for this to be natural inX? Rule (◦) says that for each tripleX,Y,Z∈Cthere is a function
◦: (X`Y) × (Y `Z) → X`Z.
What would it mean for this to be natural inX,YandZ? The answer to both questions involves a generalization of natural transformations called ‘dinatural’ transformations [71].
As noted in Definition 4, a natural transformationα:F ⇒ Gbetween two functors F,G:C → Dmakes certain squares inDcommute. If in factC=C1op×C2,then we actually obtain commuting cubes inD.Namely,
the natural transformation αassigns to each object (X1,X2) a morphism αX1,X2 such that for any morphism (f1:Y1→X1,f2:X2→Y2) inC, this cube commutes:
G(Y1,X2) G(Y1,Y2) F(Y1,X2) F(Y1,Y2) G(X1,X2) G(X1,Y2) F(X1,X2) F(X1,Y2) - G(1Y1,f2) ? G(f1,1Y2) ? F(f1,1X2) - F(1Y1,f2) αY1,X2 G(f1,1X2) ? ? F(f1,1Y2) αY1,Y2 G(1X1,f2) - - F(1X1,f2) αX1,X2 αX1,Y2
IfC1 =C2,we can choose a single objectXand a single morphism f:X → Y and use it in both slots. As
shown in Figure 1, there are then two paths from one corner of the cube to the antipodal corner that only involve
αfor repeated arguments: that is,αX,X andαY,Y, but notαX,YorαY,X. These paths give a commuting hexagon.
This motivates the following: